We now discuss the connection between CBSs and CIMs.
Using Proposition 4.1 of [1]}, a CIM can be rewritten as a CBS as
\( X_\ell = C(\xi _{\ell },\xi _{\ell -1},\xi _{\ell -2},\dots ) \text{ with } \alpha _\ell = \beta _0 \mathcal {B}^{\ell -1}.\) 
